Integral Structures on H-type Lie Algebras

2001/01/28 by G. Crandall, G. Douglas Crandall, Crandall, G. +2 · 2 citations
Mathematics · #22E25 #22E60 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:22E25 #msc:22E60

paper · pdf · doi:10.48550/arxiv.math/0101230

13 pages, LaTeX

arxiv created 2001/01/28 · openalex publication_date 2001/01/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal'cev criterion that all simply connected H-type Lie groups contain cocompact lattices. Since the Campbell-Hausdorff formula is very simple for two-step nilpotent Lie groups we can actually avoid invoking the Mal'cev criterion and exhibit our lattices in an explicit way. As an application, we calculate the isoperimetric dimensions of H-type groups.

Cited by

• An explicit rational structure for real semisimple Lie algebras
• Complete classification of H-type algebras: I

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